Vibration resistant interferometry

ABSTRACT

Scanning interferometry data for a test object is provided, the data typically including intensity values for each of multiple scan positions for each of different spatial locations of the test object. The intensity values for each spatial location define an interference signal for the spatial location, and the intensity values for a common scan position define a data set for that scan position. Scan values are provided for each scan position, in which scan value increments between various scan values can be non-uniform. Information is determined about the test object based on the scanning interferometry data and scan values. The determination includes transforming at least some of the interference signals into a frequency domain with respect to the scan values.

RELATED APPLICATION

The present application claims the benefit of U.S. provisionalapplication No. 60/564,477, filed Apr. 22, 2004, and claims priorityunder 35 U.S.C. §119(a) to Taiwanese application no. 94112683, filedApr. 21, 2005, both of which applications are incorporated herein intheir entireties.

FIELD OF THE INVENTION

The invention relates to scanning interferometry methods as well as torelated systems.

BACKGROUND

Scanning white light interferometry (SWLI) is an optical profilingprocedure by which a nominally equal path interferometer is illuminatedby a broad band source (typically but not limited to white light) whileone leg of the interferometer is axially moved (scanned) through theequal path condition (the condition when the optical path length of thetwo legs of the interferometer are equal). The interference signals arecaptured by an optical sensor, typically a camera during the scan.Interference fringes only occur in the neighborhood of the equal pathcondition, providing a signal to compute the relative height of thevarious image points of the surface illuminated by one leg to thecorresponding image point in the other leg. All SWLI methods typicallyassume a constant scanning motion (i.e. constant velocity). If thescanning motion is not uniform, errors in the measured surface profileoccur.

Unfortunately, it is often the case that the scanning motion in SWLI isnot uniform. This can occur due to nonlinear motions of the scanningmechanism, or through vibrations that act on each interferometer legdifferently.

SUMMARY

Methods and systems are disclosed that provide a low-cost way to reducethe sensitivity of surface profile measurements using SWLI to vibrationand/or nonlinear scans. They include a generalized Fourier domain SWLIprocessing technique called GFDA that can handle arbitrary SWLI scanmotion provided the scan motion itself is known.

In some embodiments, the scan motion is determined using the SWLI dataitself by measuring the phase shift increments between camera frames(e.g., data sets) via a spatial carrier fringe technique. In otherembodiments, the scan motion is determined by measuring the phase shiftincrements between camera frames via an additional inline displacementmeasuring sensor. In further embodiments, the scan motion is determinedby measuring the phase shift increments between camera frames via anadditional external displacement measuring sensor.

In general, in one aspect, the invention features a method including:(i) providing scanning interferometry data for a test object, the dataincluding intensity values at each of multiple scan positions for eachof different spatial locations of the test object; (ii) providing valuesfor the scan positions; and (iii) determining information about the testobject based on the scanning interfometry data at each of the differentspatial locations and the scan positions.

Embodiments of the method may include any of the following features.

Determining the information about the test object may includetransforming the scanning interferometry data at each of the differentspatial locations into a spatial frequency domain with respect to thevalues for the scan positions.

The scan positions may vary nonlinearly with time.

A light source having a coherence length may be used to generate thescanning interferometry data, and the scan positions may span a rangelarger than the coherence length of the light source.

A light source having a center frequency may be used to generate thescanning interferometry data, and the light source may have a spectralwidth greater than 2% of the center frequency.

Providing the values for the scan positions may include determining thescan positions.

The scan positions may be determined from the intensity values of thescanning interferometry data over at least some of the multiple spatiallocations. For example, determining the scan positions may includetransforming the intensity values at the different spatial locationsinto a spatial frequency domain for each of at least some of the scanpositions.

Determining the scan positions may include using an inline displacementsensor.

Determining the scan positions may include using an externaldisplacement sensor (e.g., a displacement measuring interferometer).

In some embodiments, a method includes providing scanning interferometrydata for a test object, the data comprising intensity values for each ofmultiple scan positions for each of different spatial locations of thetest object, the intensity values for each spatial location defining aninterference signal for the spatial location, the intensity values for acommon scan position defining a data set for that scan position. A scanvalue is provided for each scan position. Increments between the scanvalues are non-uniform. At least some of the interference signals aretransformed into a frequency domain with respect to the scan values.Information about the test object is determined based on the transformedinterference signals.

Transforming at least some of the interference signals into a frequencydomain with respect to the scan values can include transforming the atleast some interference signals over a range of scan positions greaterthan a coherence length of a light source used to generate the scanninginterferometry data.

The scanning interferometry data can be acquired using a light sourcehaving a center frequency and the light source has a spectral widthgreater than 2% of the center frequency.

In some embodiments, the scan values are determined based on thescanning interferometry data and an initial estimate of values for thescan positions.

Determining the scan positions can include determining the scanpositions based on a property of light that has passed through an opticused to receive light reflected from the test object. An in-line sensormay be used to measure the scan positions.

Determining the scan positions can include determining the scanpositions based on a property of light that has not passed through anoptic used to receive light reflected from the test object. An externalsensor may be used to measure the scan positions. The external sensormay be a displacement measuring interferometer.

In some embodiments, a method includes providing scanning interferometrydata for a test object, the data comprising an intensity valuecorresponding to each of different spatial locations of the test objectfor each of multiple scan positions. Information related to the spatiallocations of the test object is determined based on the scanninginterferometry data. A scan value is determined for one of the multiplescan positions based on a relationship between the intensities of thatscan position and the information related to the spatial locations towhich the intensities of that scan position correspond. A scan value maybe determined for each of additional ones of the multiple scanpositions.

The information related to the spatial locations of the test object maybe determined on the further basis of an initial scan value for each ofthe scan positions.

The scan position for which the scan value is determined may be a firstscan position spaced apart from a second scan position by a scanposition increment and determining a scan value for the first scanposition can include determining a scan value increment for the scanposition increment based on (a) the relationship between the intensitiesof the first scan position and the information related to the spatiallocations to which the intensities of the first scan position correspondand (b) a relationship between the intensities of the second scanposition and the information related to the spatial locations to whichthe intensities of the second scan position correspond.

Determining the scan value for the scan position can includetransforming the intensities of the scan position into a frequencydomain with respect to the information related to the spatial locationsto which the intensities of the scan position correspond. Thetransformation may be a one-dimensional transformation.

Determining the scan value for the scan position can include determininga phase of a fundamental frequency of an oscillation of the intensitiesof the scan position with respect to the information related to thespatial locations to which the intensities of the scan positioncorrespond. Determining the phase of the fundamental frequency caninclude using information about a phase of a frequency of at least asecond oscillation of the intensities of the scan position with respectto the information related to the spatial locations to which theintensities of the scan position correspond. The frequency may be atleast 3 times the fundamental frequency. Information from additional,higher frequencies, may also be used.

In some embodiments, a method includes providing scanning interferometrydata for a test object. The data includes an intensity valuecorresponding to each of different spatial locations of the test objectfor each of multiple scan positions. In formation related to the spatiallocation of the test object is determined based on the scanninginterferometry data. A scan value increment between a pair of the scanpositions is determined based on (a) a relationship between intensitiesof a first scan position of the pair and the information related to thespatial locations to which the intensities of the first scan positioncorrespond and (b) a relationship between the intensities of a secondscan position of the pair and the information related to the spatiallocations to which the intensities of the second scan positioncorrespond.

The information about the test object may be determined on the basis ofboth the scanning interferometry data and initial scan values for thescan positions (e.g., by transforming interference signals of thescanning interferometry data into a frequency domain with respect to theinitial scan values).

The pair of scan positions may be a first pair of scan positions and themethod may further include repeating the determining a scan valueincrement to determine a scan value increment between other pairs of thescan positions. A scan value increment may be determined between allpairs of successive scan positions. The scan value increments may benon-uniform.

The method may include determining information about the test objectbased on the scanning interferometry data and the scan value increments(e.g., by transformation of interference signals of the scanninginterferometry data with respect to the scan value increments).

Determining a scan value increment can include fitting a first functionto the at least some intensities of the first one of the scan positionsand the information related to the spatial locations corresponding tothe at least some intensities and fitting a second function to the atleast some intensities of the second one of the scan positions and theinformation related to the spatial locations corresponding to the atleast some intensities. The scan value increments can be determinedbased on fitted parameters of the first and second functions.

Determining a scan value increment can include transforming at leastsome of the intensity values of the first scan position into a frequencydomain with respect to the information related to the spatial locationscorresponding to the intensity values and transforming at least some ofthe intensity values of the second scan position into a frequency domainwith respect to the information related to the spatial locationscorresponding to the intensity values.

Determining a scan value increment between a pair of scan positions caninclude determining an offset between (a) a fundamental frequency of anoscillation of the at least some intensities of the first one of thescan positions with respect to the information related to the spatiallocations corresponding to the at least some intensity values of thefirst one of the scan positions and (b) a fundamental frequency of anoscillation of the at least some intensities of the second one of thescan positions with respect to the information related to the spatiallocations corresponding to the intensity values of the second one of thescan positions. Determining the offset can include using informationfrom an oscillation of at least about 3 times the fundamental frequencyof the at least some intensities of the first one of the scan positionswith respect to the information related to the spatial locationscorresponding to the at least some intensity values of the first one ofthe scan positions and using information from an oscillation of about 3times the fundamental frequency of the at least some intensities of thefirst one of the scan positions with respect to the information relatedto the spatial locations corresponding to the at least some intensityvalues of the second one of the scan positions.

In some embodiments, a method comprises providing scanninginterferometry data for a test object, the data comprising intensityvalues for each of multiple scan positions for each of different spatiallocations of the test object, the intensity values for each spatiallocation defining an interference signal for the spatial location, theintensity values for a common scan position defining a data set for thatscan position. Information about a test object is determined bytransforming at least some of the interference signals into a frequencydomain with respect to initial values for the scan positions. Based onthe scanning interferometry data and the information about the testobject, a scan value is determined for each of at least some of the scanpositions. The scan values may be scan value increments between pairs ofscan positions.

Information about the test object may be determined based on thescanning interferometry data and the scan values for the scan positions.Determining information about the test object can include transformingat least some of the interference signals into a frequency domain withrespect to the scan values.

Scan value increments between the scan values may be non-uniform.

Transforming at least some of the interference signals into a frequencydomain with respect to the scan positions can include transforming theat least some interference signals over a range of scan positionsgreater than a coherence length of a light source used to generate thescanning interferometry data.

The scanning interferometry data can be acquired using a light sourcehaving a center frequency and the light source has a spectral widthgreater than 2% of the center frequency.

In another embodiment, a system includes an interferometer configured toprovide scanning interferometry data for a test object, the datacomprising intensity values for each of multiple scan positions for eachof different spatial locations of the test object, the intensity valuesfor each spatial location defining an interference signal for thespatial location, the intensity values for a common scan positiondefining a data set for that scan position and a processor configured totransform at least some of the interference signals into a frequencydomain with respect to initial values for the scan positions anddetermine, based on at least some of the transformed interferencesignals, a respective scan value for each of at least some of the scanpositions.

In another embodiment, a system includes an interferometer configured toprovide scanning interferometry data for a test object, the datacomprising intensity values for each of multiple scan positions for eachof different spatial locations of the test object, the intensity valuesfor each spatial location defining an interference signal for thespatial location, the intensity values for a common scan positiondefining a data set for that scan position and a processor configured totransform the interference signals for each of different spatiallocations into a spatial frequency domain with respect to scan values,each scan value corresponding to a scan position. A difference betweendifferent scan values is non-uniform.

The system may include a sensor with the processor configured todetermine the second scan values based on data received from the sensor.

The processor may be configured to determine scan values based on atleast some intensity values of the scanning interferometry data and,optionally, initial scan values for the scan positions.

The processor may be configured to determine information about the testobject based on the intensity values transformed with respect to thesecond scan values.

In some embodiments, a system includes an interferometer configured toprovide scanning interferometry data for a test object, the datacomprising an intensity value corresponding to each of different spatiallocations of the test object for each of multiple scan positions and aprocessor configured to determine information related to the spatiallocations of the test object based on the scanning interferometry dataand determine a scan value for a scan position based on a relationshipbetween the intensities of the scan position and the information relatedto the spatial locations to which the intensities of the scan positioncorrespond. The processor may be configured to determine informationabout the test object based on information related to the scan value.

In some embodiments, a method includes providing scanning interferometrydata for a test object. The data include intensity values at each ofmultiple scan positions for each of different spatial locations of thetest object. Scan values are provided for the scan positions.Information about the test object is determined based on the scanninginterferometry data at each of the different spatial locations and thescan positions.

Determining the information about the test object can includetransforming the scanning interferometry data at each of the differentspatial locations into a spatial frequency domain with respect to thescan values for the scan positions. The scan positions may varynonlinearly with time.

In some embodiments, a light source having a coherence length is used togenerate the scanning interferometry data. The scan positions may span arange larger than the coherence length of the light source.

In some embodiments, a light source having a center frequency is used togenerate the scanning interferometry data. The light source can have aspectral width greater than 2% of the center frequency.

Providing the scan values for the scan positions may include determiningthe scan positions. In some embodiments, the scan positions aredetermined from the intensity values of the scanning interferometry dataover at least some of the multiple spatial locations. For example,determining the scan positions can include transforming the intensityvalues at the different spatial locations into a spatial frequencydomain for each of at least some of the scan positions.

In some embodiments, determining the scan positions includes using aninline displacement sensor.

In some embodiments, determining the scan positions includes using anexternal displacement sensor. The external displacement sensor may be adisplacement measuring interferometer.

In another aspect, the invention features a system including: (i) ascanning interferometer configured to provide scanning interferometrydata for a test object, the data including intensity values at each ofmultiple scan positions for each of different spatial locations of thetest object; (ii) means for determining the scan positions; and (iii) anelectronic controller configured to transform the scanninginterferometry data at each of the different spatial locations into aspatial frequency domain with respect to the measured scan positions,and determine information about the test object based on the transformedscanning interfometry data at each of the different spatial locations.

In some embodiments, the system includes a scanning interferometerconfigured to provide scanning interferometry data for a test object.The data include intensity values at each of multiple scan positions foreach of different spatial locations of the test object. The system alsoincludes means for determining the scan positions and an electroniccontroller configured to transform the scanning interferometry data ateach of the different spatial locations into a spatial frequency domainwith respect to the measured scan positions and determine informationabout the test object based on the transformed scanning interferometrydata at each of the different spatial locations.

The systems may further include features corresponding to the methoddescribed above.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs.

Other features, objects, and advantages of the invention will beapparent from the following detailed description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows an embodiment of a scanning interferometry system.

FIG. 2 shows an interference signal corresponding to a spatial locationof a test object.

FIG. 3 illustrates a schematic side view of a test object andinterference signals corresponding to different test object spatiallocations.

FIGS. 4 a-4 c illustrates intensity values obtained for common scanpositions from different spatial locations of the test object of FIG. 3.

FIG. 5 illustrates the two-dimensional spatial transform of theintensity values of FIG. 4 b.

FIG. 6 illustrates a two-dimensional Fourier filter.

FIG. 7 illustrates the absolute value of inverse transform of theproduct of the spatial transform of FIG. 5 and the filter of FIG. 6.

FIG. 8 illustrates the intensity values for a common scan positionobtained for a common scan position from a spherical test object.

FIG. 9 illustrates a region of intensity values (e.g., pixels) of FIG. 8for which the intensity values are within a range R of the maxima of theinterference signals corresponding to the intensity values.

FIG. 10 illustrates (a) the relationship between intensity values withinFIG. 8 as shown by FIG. 9 and information about the test object asexpressed in scan position units and (b) the corresponding relationshipfor intensity values obtained at a scan position adjacent the commonscan position of FIG. 8.

FIG. 11 illustrates an embodiment of an interferometer having a sensorfor measuring scan position increments.

FIG. 12 illustrates an embodiment of an interferometer having a sensorfor measuring scan position increments.

FIG. 13 illustrates an embodiment of an interferometer having a sensorfor measuring scan position increments.

FIG. 14 illustrates an embodiment of an interferometer having a sensorfor measuring scan position increments.

FIG. 15 illustrates an embodiment of an interferometer having a sensorfor measuring scan position increments.

FIG. 16 a illustrates a topography of a flat test object in the presenceof a 1 HZ, 30 nanometer vibration determined using frequency domainanalysis of scanning interferometry data based on scan positions of thedata.

FIG. 16 b illustrates a topography of the flat test object of FIG. 16 bdetermined using generalized frequency domain analysis of scanninginterferometry data based on scan values for the scan positions.

FIG. 16 c illustrates optical path difference variations represented bythe increments between the scan values used to determine the topographyof FIG. 16 b.

FIG. 17 a illustrates a topography of a flat test object in the presenceof a 3 HZ, 30 nanometer vibration determined using frequency domainanalysis of scanning interferometry data based on scan positions of thedata.

FIG. 17 b illustrates a topography of the flat test object of FIG. 17 bdetermined using generalized frequency domain analysis of scanninginterferometry data based on scan values for the scan positions.

FIG. 17 c illustrates optical path difference variations represented bythe increments between the scan values used to determine the topographyof FIG. 17 b.

FIG. 18 illustrates the topography of a spherical object (with thespherical shape component removed) as determined from each of 4independent sets of scanning interferometry data based on frequencydomain analysis using the scan positions.

FIG. 19 illustrates the topography of the spherical object of FIG. 18(with the spherical shape component removed) as determined from each ofthe 4 independent sets of scanning interferometry data based ongeneralized frequency domain analysis using scan values for the scanpositions.

FIG. 20 illustrates the non-uniformity of the scan value incrementsbetween the scan values used to determine the topographies of FIG. 19.

FIGS. 21 and 22 are flow diagrams of example processes.

DETAILED DESCRIPTION

Scanning interferometry data are generally obtained by detecting theintensity of interference between light (e.g., ultraviolet light,visible light, or infrared light) reflected from a test object and lightreflected from a reference object. The intensity of the interferencedepends on an optical path difference (OPD) between the light reflectedfrom the test object and light reflected from the reference object. Foreach spatial location of the test object (e.g., for each of differentlocations about a surface of the test object), the OPD depends uponproperties of the spatial location (e.g., the height of the spatiallocation).

Typically, scanning interferometry data are obtained by using a scanmechanism to vary the OPD between the light reflected from the testobject and the light reflected from the reference object. The scanningmechanism generally moves at least one of the test and reference objectsthrough multiple scan positions along a scan dimension (e.g.,continuously or in discrete steps). Each scan position corresponds to adifferent OPD between the test and reference objects. For each scanposition, a detector detects the intensity of interference for each ofmultiple spatial locations of the test object. The intensities can beused to determine information about the test object.

Information about a test object that can be determined from scanninginterferometry data includes, for example, information related to aspatial property (e.g., a height such as relative height) of one or morespatial locations of the test object. Examples of information related toa spatial property (e.g., height) of a test object include an OPD (e.g.,an OPD corresponding to a spatial location of the test object) and aphase of interference (e.g., a phase of interference between lightreflected from a spatial location of the test object and light reflectedfrom the reference object).

Determining information about the test object typically includes the useof initial scan values for the scan positions. Use of the initial scanvalues assumes ideal scan mechanism motion (e.g., uniform and precisemotion) along the scan dimension. Hence, scan value increments betweenthe initial scan values are uniform along the scan dimension. Inpractice, scan mechanism motion tends to have some degree ofnon-ideality (e.g., non-uniformity and/or imprecision). One source ofnon-ideal scan mechanism motion includes vibrations (e.g., from aircurrents and/or nearby machinery), which can cause one or both of thetest and reference objects to move (e.g., vibrate) with respect to theother. Another source of non-ideal scan mechanism motion includesnon-uniform (e.g., non-linear) motion of the scan mechanism along thescan dimension. The use of initial scan values that incorporate theassumption of ideal scan mechanism motion can introduce artifacts (e.g.,errors) into the information about the test object.

As described herein, scan values can be provided for each of at leastsome (e.g., all) of the scan positions of scanning interferometry datafor a test object. In general, the scan values can accommodate fornon-ideal motion of the scan mechanism. At least some (e.g., most orall) of the scan value increments between different scan values may bedifferent (e.g., the scan value increments may be non-uniform).

Information about the test object can be determined based on thescanning interferometry data and the scan values. Typically, theinformation about the test object determined based on the scan valuescontains fewer artifacts (e.g., fewer and smaller magnitude errors) withrespect to the test object than information determined based on initialscan values that incorporate the assumption of ideal scan mechanismmotion.

Scan values provided for the scan positions can be expressed in manydifferent units (e.g., units of phase, distance, and/or OPD). The scanvalues can be absolute (e.g., the scan values can relate to an absolute(e.g., total) OPD between test and reference objects) or relative (e.g.,the scan values can be scan value increments related to an incrementalOPD change between the test and reference objects for different (e.g.,successive) scan positions). Scan values can be expressed as scan valueincrements related to increments between absolute scan values.

In some embodiments, the information about the test object is determinedbased on interference signals corresponding to each of at least some ofdifferent test object spatial locations. The determination can includetransforming (e.g., by Fourier methods) the interference signals into afrequency domain with respect to the scan values. Such methods cangeneralize transform methods to account for non-ideal scan mechanismmotion.

In some embodiments, scan values (e.g., relative scan values, such asscan value increments) for the scan positions (e.g., for the scanposition increment between one or more scan positions) are determinedusing the scanning interferometry data themselves. For example,determining a scan value for a scan position can include determininginformation related to multiple spatial locations of a test object basedon scanning interferometry data and determining a scan value for thescan position based on a relationship between the intensities of thescan position and the information related to the spatial locations towhich the intensities of the scan position correspond. Typically, theinformation related to the spatial locations is related to the heightsof the spatial locations.

In some embodiments, scan values are obtained for each scan positionwhile obtaining the scanning interferometry data (e.g., by measuring thenominal OPD corresponding to the scan position and/or by measuring thedeviation from ideality of the OPD created by the scan mechanismmotion). For example, the scan values can be measured with an externaldisplacement sensor (e.g., a sidecar sensor) mounted, for example,externally to an interferometer (e.g., to a microscope thereof) andrigidly attached either to a scanner apparatus of the interferometer orto the microscope body (e.g., to its gantry). The sensor independentlymonitors the distance between itself and the test object. Depending onwhat is monitored, the scan values can be used either as a directmeasure of the scan positions or as a measurement of the deviation fromexpected scan positions.

Referring now to FIG. 1, an exemplary interferometer system 50 forobtaining interference signals includes an interferometer 51 and aprocessor 52 (e.g., an automated computer control system). Themeasurement system 50 is operable to obtain scanning interferometry dataof spatial locations of a test object 53.

Measurement system 50 includes a light source 54, a first focusing optic(e.g., one or more lenses) 56, a beam splitting element 57, a secondfocusing optic 62, a reference object 58, a third focusing optic 60, anda detector 59. Light source emits 54 emits spectrally-broadband light(e.g., white light), which illuminates a diffusing screen 55. Firstfocusing optic 56 collects light from screen 55 and transmits collimatedlight to beam-splitting element 57, which splits the collimated lightinto first and second portions. A first portion of the collimated lightis received by second focusing optic 62, which focuses the first portionof the light onto reference object 58. Light reflected from thereference object is received by second focusing optic 62, whichtransmits collimated light reflected by the reference object 58 back tobeam-splitting element 57. Beam-splitting element 57 directs the secondportion of the collimated light to third focusing optic 60, whichfocuses the light onto test object 53. Light reflected from test object53 is received by third focusing optic 60, which transmits collimatedlight reflected by test object 53 back to beam-splitting element 57.Beam-splitting element 57 combines light reflected from reference object58 and test object 53 and directs the combined light to a fourthfocusing optic 61, which focuses the combined light to a detector 59.

Detector 59 is typically a multidimensional detector (e.g., a chargecoupled device (CCD) or charge injection device (CID)) having aplurality of detector elements (e.g., pixels) arranged in one or moredimensions (e.g., two dimensions). Optics 60 and 61 focus lightreflected from test object 53 onto detector 59 so that each detectorelement of detector 59 receives light reflected from a correspondingspatial location (e.g., a point or other small region) of test object53. Light reflected from respective spatial locations of test object 53and light reflected from reference object 58 interferes at detector 59.Each detector element produces a detector signal related to theintensity of the interfering light.

System 50 is configured to measure interference signals related tospatial locations of test object 53. Typically, system 50 creates an OPDbetween light reflected from reference object 58 and light reflectedfrom test object 53. For example, test object 53 can be displacedthrough a number of scan positions along a scan dimension axis by a scanmechanism (e.g., an electromechanical transducer 63 (e.g., apiezoelectric transducer (PZT)), and associated drive electronics 64)controlled by computer 52. In some embodiments, a scan positionincrement between successive scan positions is at least about λ/15(e.g., at least about λ/12, at least about λ/10), where λ is a meanwavelength of the light detected at each pixel.

For each scan position, detector 59 outputs an intensity value (e.g.,the intensity detected by a given detector element) for each of multipledifferent spatial locations of the test object. Taken along the scandimension, the intensity values for each spatial location define aninterference signal corresponding to the spatial location. The intensityvalues corresponding to a common scan position define a data set (e.g.,an interferogram) for that scan position.

Referring to FIG. 2, an exemplary interference signal 75 includes anintensity value 77 j for each of N scan positions 79 _(i) along a scandimension axis 81. The intensity values 77 j of interference signal 75correspond to a single spatial location of a test object. As discussedabove, scan position increments between successive scan positions aretypically uniform.

Intensity values 77 j map out a number of oscillations (e.g., fringes),which decay on either side of a maximum according to a low coherenceenvelope 83, which does not expressly appear in such interferencesignals but is shown for clarity. The width of coherence envelope 83corresponds generally to the coherence length of the detected light.Among the factors that determine the coherence length are temporalcoherence phenomena related to, for example, the spectral bandwidth ofthe light, and spatial coherence phenomena related to, for example, therange of angles of incidence of light illuminating the test object. Ascan be seen from FIG. 2, interference signal 75 results from detectingintensity values over a range of scan positions 79 j that is greaterthan about ¾ of the width of the coherence envelope. In someembodiments, the intensity values are detected over a range of scanpositions that is greater than the width of the coherence envelope and,therefore, greater than the coherence length of the detected light.

In some embodiments, the motion of the scan mechanism is continuousalong the scan dimension. In the absence of non-ideal scan mechanismmotion, the time-dependent interference intensity s(t) detected by agiven pixel can be described as:s(t)=I ₀{1+V cos[Φ+φ(t)]}  (Eq. 1)where I₀ is the average interference intensity, V is the contrast of theinterference intensity variation (e.g., the fringe contrast), φ(t) isthe change in the interference phase introduced by the scanningmechanism motion along the scan dimension axis, and Φ is theinterference starting phase, which is related to information about thetest object spatial location corresponding to the given pixel. Ingeneral, φ(t) is given by φ(t)=2πv₀t, where v₀ is the fundamentalfrequency of oscillation along the scan dimension axis, given byv₀=2v_(r)/λ, where λ is the mean wavelength of the detected light.

In the presence of vibrations, the time-dependent interference intensitydetected at each pixel can be described as:s(t)=I ₀{1+V cos[Φ+φ(t)+r cos(2πv _(v) t+α)]}  (Eq.2)where v_(v) is the frequency of the intensity disturbance, α is startingphase of the intensity disturbance, and r is the disturbance amplitude.As discussed above, the presence of the vibrations and other non-idealscan mechanism motion can introduce errors when information about theobject is determined from interference signals based on initial scanvalues (which neglect non-ideal scan mechanism movement). The errorsintroduced by the vibrational disturbance contaminate not only theinterference signals along the scan dimension axis but the intensityvalues detected for a common scan position. Although not reflected byEq. 2, errors introduced by non-uniform scan mechanism motion alsoperturb both the interference signals and the intensity data for eachscan position.

Typically, the scan mechanism of system 50 moves continuously throughthe scan positions and the system acquires intensity values for eachscan position by integrating the intensity detected by each pixel for atime Δt centered about each scan position. The rate of data setacquisition is given by (Δt)⁻¹. In general, (Δt)⁻¹ is smaller thanv_(v). For example, in some embodiments, the ratio (Δt)⁻¹/v_(v) is about0.75 or less (e.g., about 0.5 or less, about 0.3 or less, about 0.2 orless, about 0.15 or less). In an exemplary embodiment, the ratio(Δt)⁻¹/v_(v) is about 0.1.

Intensity values are typically obtained for each of at least about 64(e.g., at least about 128) scan positions. In some embodiments,intensity values are obtained at each of about 1028 scan positions orfewer (e.g., about 514 scan positions or fewer, about 256 scan positionsor fewer).

In general, detector 59 includes at least about 4,000 (e.g., at leastabout 256,000, at lest about 1,024,000) detector elements. The scanninginterferometry data typically include a number of interference signalsequal to the number of detector elements and corresponding to the samenumber of test object spatial locations. Processor 50 can process thescanning interferometry data (e.g., using methods described herein) todetermine, for example, information about the test object.

Light source 54 is typically a spectrally-broadband source (e.g., awhite-light lamp) or a source including a plurality of differentwavelengths (e.g., a plurality of light emitting diodes each producing adifferent wavelength). In some embodiments, the light reflected from thetest object 53 and reference object 58 has a center frequency and aspectral width greater than 2% of the center frequency. In someembodiments, the scanning interferometry data are scanning white lightinterferometry (SWLI) data.

As an alternative or in combination with a broadband source, the lightsource 54 can include a narrow band or quasi-monochromatic source,typically having a high numerical aperture. A low coherence interferencesignal can be obtained using a monochromatic source in combination witha high numerical aperture, e.g., the coherence length may be on theorder of a few microns or less.

In some embodiments, scanning interferometry data are acquired bychanging the OPD over a range about equal to or smaller than a coherencelength of the light detected after reflection from test object 53 andreference object 58. For example, light source 54 may be a laser havinga line width much narrower and a coherence length much longer than thespectral width and coherence length of a white light lamp.

In some embodiments, reference object 58 is optically flat and includesonly a single reflecting surface. For example, reference object 58 canbe a reference mirror. In some embodiments, reference object 58 exhibitsa three-dimensional surface topography.

We turn next to methods for determining information about a test object.Before discussing scan values that accommodate non-ideal scan mechanismmotion, we discuss methods for determining information about a testobject based on initial scan values (e.g., initial scan values thatassume ideal scan mechanism motion). An exemplary method for determininginformation about a test object includes transforming the interferencesignals of the scanning interferometry data into a frequency domain withrespect to initial estimates of the scan positions. For example, asuitable transform-based method referred to as frequency domain analysis(FDA) is described in U.S. Pat. No. 5,398,113 to de Groot, the contentsof which patent is incorporated herein by reference.

In general, FDA includes determining the transform of the interferencesignal obtained by varying the OPD along a scan dimension. The height Zof each test object spatial location is determined from the phase slopeof the transformed interference signal of the corresponding pixel givenby:

$\begin{matrix}{Z = \frac{\mathbb{d}\varphi}{\mathbb{d}k}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$where φ is the phase of the transformed interference signal and k is theconjugate variable to scanning dimension coordinate of the untransformedinterference signals. Typically, k is a wavenumber given by k=4π/λ,where λ is a wavelength of light of the interference signal. Thedefinition of the wavenumber k incorporates a factor of 2 increase inOPD for each incremental increase in scan value. The height Z (e.g., thephase slope) can be determined by, for example, fitting a plot of phaseφ vs. wavenumber k for several highest power spectral components. Ahigher precision estimate of object height can be based on the phase ofthe transformed interference signal at a high power wavenumber and usingthe phase slope Z to unwrap any 2π ambiguity.

The foregoing analysis uses initial scan values that incorporate theassumption of ideal scan mechanism motion (e.g., increments between theinitial scan values are typically uniform). As discussed above, however,non-ideal scan mechanism motion can cause the actual scan positions todeviate from those expected based on operation of the interferometer.

We next turn to a discussion of a method for determining informationabout a test object based on scan values z_(j) that accommodatenon-ideal scan mechanism motion (e.g., vibration and/or non-uniform scanmechanism motion). Here, we assume that these scan values z_(j) can beprovided. Below, we discuss methods for providing (e.g., determining)the scan values z_(j).

In some embodiments, the method for determining information based on thescan values z_(j) includes a generalized transformation of one or moreinterference signals of scanning interferometry data with respect to thescan values z_(j). Information determined on the basis of thegeneralized transformation typically includes fewer artifacts (e.g.,errors) caused by non-ideal scan mechanism motion as compared toinformation determined on the basis of an initial estimate of scanvalues that neglect non-ideal motion.

In general, the scan value z_(j) for the jth scan position is related tothe scan value increments ∂z_(j)=z_(j)−z_(j−1) between the scan valuesby:

$\begin{matrix}{{z_{j} = {\overset{\_}{z} + {\sum\limits_{i = 0}^{j - 1}{\delta\; z_{i}}}}},} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$where z corresponds to a scan position offset, which is typically aboutzero (e.g., z generally corresponds to an OPD between the test andmeasurement objects of about zero).

Typically, the scan value increments δz_(j) between different scanvalues are different (e.g., non-uniform) along the scan dimension axis.For example, the scan value increments can include a random (oroscillatory) component that accounts for scan position errors introducedby vibrations. The scan value increments can also include asystematically changing component (e.g., a component that increases ordecreases) along at least a portion (e.g., all) of the scan dimensionaxis that accounts for scan position errors introduced by non-uniformscan mechanism motion. Such systematic components can be linear ornon-linear.

A generalized transform for transforming an interference signal into afrequency domain with respect to the scan values z_(j) is given by:

$\begin{matrix}{{P(k)} = {\sum\limits_{j}{p_{j}{\exp\left( {{\mathbb{i}}\;{k\left( {z_{j} - \overset{\_}{z}} \right)}} \right)}{\partial z_{j}}}}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$where P(k) is the interference signal transformed based on the scanvalues zj, p_(j) is intensity value of the interference signal for thejth scan value zj, and k is defined as above.

Information about each test object spatial location is typicallydetermined based on the phase of at least one wavenumber of thetransformed data. The phase φ(k) of the interference signal atwavenumber k is given by:φ(k)=arg(P(k))=a tan(Im(P(k))/Re(P(k)))  (Eq.6),where Im(P(k)) is the imaginary portion of the transformed interferencesignal at wavenumber k and Re(P(k) is real portion of the transformedinterference signal at wavenumber k. The height of each test objectspatial location can be determined, for example, as described abovebased on a fit to a phase φ vs. wavenumber k plot of several high powerspectral components. A higher precision estimate of object height can bebased on the phase of the transformed interference signal at a highpower wavenumber and using the phase slope to unwrap any 2π ambiguity.

Typically, information about a test object spatial locationcorresponding to the transformed interference signal is determined basedon one or more wavenumbers of the transformed interference signaladjacent (e.g., coincident and/or straddling) the mean wavenumber k₀,which is related to the mean wavelength λ₀ of the light of theinterference signal by k₀=4π/λ₀. Wavenumbers adjacent k₀ typically havea spectral amplitude |P(k)| greater than a threshold, which can bechosen as desired (e.g., to reduce the impact of noise on the result).

The foregoing discussion assumes that scan values z_(j) can be providedfor at least some (e.g., all) scan positions. We turn now to methods fordetermining the scan values z_(j) (e.g., the scan value increments∂z_(j)) that accommodate non-ideal scan mechanism motion. As discussedwith respect to Eq. 2, non-ideal scan motion perturb both theinterference signals along the scan dimension axis and the intensityvalues observed for each common scan position. Hence, the intensityvalues contain information about the non-ideal scan mechanism motion. Insome embodiments, the scan values z_(j) are determined from scanninginterferometry data of a test object.

Typically, the scanning interferometry data include an intensity valuefor each of multiple spatial locations of the test object for each ofmultiple scan positions (e.g., the data include an interference signalfor each of multiple test object spatial locations and a data set foreach of multiple scan positions). In some embodiments, the scanninginterferometry data are obtained with the test and reference objectstilted with respect to one another to introduce spatial carrier fringesin the intensity values obtained for each scan position. FIG. 3 shows aschematic side view of a test object 100 and the relative position ofinterference signals resulting from each of 4 spatial locations withrespect to a scan dimension axis 105. For example, at scan position T2along scan dimension axis 105, interference from test object spatiallocations 1 and 4 are at a maximum. FIGS. 4 a-4 c graphically shows theintensity values of a data set 101 for scan position T1, of a data set102 for scan position T2, and of a data set 103 for scan position T3.

The average of all the intensity values of the scanning interferometrydata is subtracted from each intensity value to reduce the DC componentin the spatial frequency domain for each data set. Eachaverage-subtracted data set is transformed (e.g., by a 2-dimensionalFourier transformation) to a spatial frequency domain. The carriersignal peak position can be determined from the location of the 1^(st)order peak in the spatial frequency domain power spectrum. For example,as seen in FIG. 5, the spatial frequency domain power spectrum obtainedfrom the transform of data set 102 shows positive and negative 1^(st)order peaks corresponding to the carrier signal peaks. Typically, thepower spectra of several data sets are averaged to reduce noise.

The spatial frequency domain power spectrum is spatially filtered toextract only one of the 1^(st) order peaks (e.g., the positive peak).FIG. 6 shows a spatial filter 112. Typically, filtering is performed inthe frequency domain as the product of the frequency domain powerspectrum and spatial filter (of course, filtering could be performed byconvolution in the spatial domain). The spatially filtered frequencydomain power spectrum is inverse transformed (e.g., by an inverseFourier transform) to obtain a complex surface map. FIG. 7 shows acomplex surface map 114 obtained by inverse transforming the product ofpower spectrum 110 and spatial filter 112. The absolute value of thecomplex surface map indicates where interference appears in thecorresponding data set (e.g., data set 102). To determine the scanincrement δzj between each pair of data sets, the absolute valuescorresponding to the pixels of the surface map are searched for valuesthat exceed a threshold, which can be selected as desired. The spatialphase at each pixel is given by the argument of the complex surface mapat each pixel. The difference between the argument for a given pixelbetween different data sets indicates the scan value increment for thosedata sets.

Scan value increments are determined by unwrapping the phase variationfor each pixel over multiple (e.g., all) data sets (e.g., within eachinterference signal) and calculating the scan value increment betweenscan positions corresponding to pairs of data sets. The increments δzjdetermined from pixels within each the data sets of each pair of datasets can be averaged together. The scan value zj for each data set canbe determined from the sum of increments δzj as discussed above. Theincrements δzj between different scan values zj are typically different(e.g., the increments can be non-uniform). The scan values thus obtainedcan be used to determine information about the surface (e.g., bytransforming the interference signals into a frequency domain withrespect to the scan values and/or by another method).

While the foregoing method for determining scan values (e.g., scanincrements δz_(j)) from scanning interferometry data has been describedas including a two-dimensional transformation of the spatialdistribution of intensities within one or more data sets, other methodsfor determining scan values from scanning interferometry data can beused. For example, in some embodiments, information about a test object(e.g., the phase of interference corresponding to one or more spatiallocations, the height of one or more spatial locations, or the OPDcorresponding to one or more spatial locations) is determined based onscanning interferometry data and, optionally, initial scan values forthe scan positions. Typically, the information about the test object isdetermined by transforming the scanning interferometry data into afrequency domain with respect to initial scan values to determineheights of various spatial locations of the test object. For intensityvalues of a common scan position, the intensity values are plotted as afunction of the information about the test object information thatcorresponds to the intensity values. For example, each intensity valuefrom a common scan position can be plotted against the height of thetest object spatial location that corresponds to that intensity value.

We now discuss the method further. The method typically includesobtaining scanning interferometry data from a test object. FIG. 8 showsan exemplary data set 165 representing the data set for the 165^(th)scan position of scanning interferometry data obtained from a sphericaltest object. Because the variation of the heights of the test objectspatial locations within the field of view is greater than the coherencelength of the detected light, fringes are only observed within a portionof the data set.

On the other hand, certain test objects (e.g., generally planar testobjects) may have a distribution of surface heights that is small withrespect to the wavelength of the detected interfering light. The numberof fringes within each data set gathered from such objects may be small(e.g., substantially less than about 1 fringe). Typically, a tilt isintroduced between such test objects and the reference object tointroduce a greater number of fringes (e.g., at least about 1 fringe, atleast about 2 fringes) within each data set. In some embodiments, thenumber of fringes created by the tilt is about 20 fringes or less (e.g.,about 10 fringes or less, about 5 fringes or less, about 2 fringes orless). Typically, the extent of tilt and the width of the coherenceenvelope of the interfering light are such that high contrast fringesare not present over an entire data set. For example, high contrastfringes may be present over about 90% of a data set or less (e.g., overabout 75% of a data set or less, over about 50% of a data set or less,over about 35% of a data set or less). Typically, the extent of tilt isinsufficient to introduce significant retrace error into informationdetermined about a test object.

The method of determining scan values z_(j) proceeds by determininginformation related to the test object based on the scanninginterferometry data and, optionally, initial values for the scanpositions. Typically, the information about the test object is obtainedby transforming the interference signals of the scanning interferometrydata into a frequency domain with respect to the initial scan values forthe scan positions (e.g., by using FDA analysis based on the initialscan values). However, other methods (e.g., analysis of the intensityvalues in the time or spatial domain of the scan positions) can be usedto determine the information about the test object.

For each intensity value of a common scan position, the informationrelated to the test object locations is typically expressed in unitsrelated to scan position. Of course, information about the test objectcan be expressed in other units (e.g., distance (e.g., nanometers) orphase (e.g., fractions of 2π)). The intensity values for a common scanposition are ranked (e.g., in ascending or descending order) based onthe information about the test object spatial locations corresponding toeach intensity value (e.g., the intensity values can be ranked in orderof the height of the test object spatial location that corresponds toeach intensity value).

For each scan position, the intensity values are searched for intensityvalues that are within a number R scan position increments of themaximum of the corresponding interference signal. In general, R is about25 or less (e.g., about 10 or less, about 7 or less, about 3) scanposition increments. For example, referring to FIG. 9, a mask 127indicates the location of a group 129 of intensity values (e.g., pixels)within data set 165. The interference signal corresponding to eachintensity value (e.g., each pixel) within group 129 has a maximum withinR=±3 scan positions of the 165^(th) scan position.

For each scan position, a scan value can be determined from therelationship between intensity values for the scan position (e.g., theintensity values for which the maximum of the corresponding interferencesignal is within ±R scan position increments (e.g., the intensity valuescorresponding to the pixels of group 129)) and information about thetest object location corresponding to each intensity value. For example,a suitable relationship is exhibited in a scatter plot of the intensityvalues against the height of the test object spatial locationcorresponding to each intensity value. FIG. 10 shows such a relationshipfor the intensity values of group 129 of the 165^(th) data set plottedas a scatter plot 130 against the heights of the test object locationsand a similar relationship for the intensity values of a correspondinggroup of the 164^(th) data set (not shown) plotted as a scatter plot 132against the heights of the test object locations. The heights areexpressed in units of scan position.

Scan values (e.g., scan value increments δz_(j) (e.g.,δz_(j)=z_(j)−z_(j−1))) for successive scan positions (e.g., between thejth and jth−1 scan positions) can be determined based on theaforementioned relationships from the intensity values of the jth dataset and the jth−1 data set. For example, the scan value increment δz₁₆₅can be determined based on the scatter plots 130,132 (e.g., bydetermining the offset between the scatter plots). The scan value z₁₆₅corresponding to the 165^(th) scan position can be determined from thesum of scan increments determined for scan positions 1-164 (e.g., usingthe scan value increments δzj between successive pairs of scan positionfor scan positions 1-164). Alternatively, the scan value for each scanposition can be determined directly from the scatter plot of intensityvalues for that scan position (e.g., from the phase of the scatterplot).

Information about the test object having reduced artifacts can bedetermined using the scan values zj corresponding to each scan positionas determined from the scanning interferometry data. In someembodiments, the determination includes transforming interferencesignals of the scanning interferometry data into a frequency domain withrespect to the scan values zj (e.g., by using generalized transformtechniques discussed herein).

In some embodiments, a scan value between a pair of scan positions isdetermined by fitting a first function to intensities for a first scanposition of the pair with respect to the information related to thecorresponding test object spatial locations (e.g., by fitting a functionto the relationship illustrated by scatter plot 130) and fitting asecond function to intensities for the second scan position of the pairwith respect to the information related to the corresponding test objectspatial locations (e.g., by fitting a function to the relationshipexpressed by scatter plot 132). Scan values (e.g., scan value incrementsδz_(j)) between the data sets of the pair are determined from theparameters of the fitted functions.

In some embodiments, the parameters are determined from a least squaresfit of first and second functions to scatter plots 130, 132. Forexample, scatter plot 130 can be fit to a function:Ui=A+B′(θ_(i))cos(θ_(i)+φ),  [7]where, the U_(i) are the intensity values of for the scan position(e.g., for the 165^(th) scan position), A is a constant, the θ_(i) arerelated to the information about the test object spatial locationcorresponding to each intensity value as determined using the scanninginterferometry data and initial scan values for the scan positions(e.g., θ_(i) can be height of the test object location corresponding toeach intensity value U_(i)), φ is related to the offset from θ_(i) ofthe scan position (e.g., the offset for the 165^(th) scan position), B′is the coherence envelope that depends on θ_(i), and i runs through allintensity values within the range determined by R. Scatter plot 132 canbe fit to a function:V _(i) =A′+B″(θ_(i))cos(θ_(i)+φ)  (Eq.8)

the V_(i) are the intensity values for the scan position (e.g., for the164th scan position), A′ is a constant, B″ is the coherence envelope, φis related to the offset from θ_(i) of the scan position (e.g., theoffset of the 164^(th) scan position). For clarity, the subscripts i areomitted in the following discussion.

For low coherence scanning interferometry data (e.g., SWLI data), thecoherence envelope (e.g., B′) can be described, for example, by:B′(θ)=Bθ ² +Cθ+D  (Eq.9).

The scatter plot data can be fit using least squares (e.g., matrix leastsquares). For example, the matrix equation for determining φ is:M·X=Y  (Eq.10),

-   -   where

$\begin{matrix}{M = \begin{bmatrix}{\sum 1} & {\sum{c\;\theta^{2}}} & {\sum{c\;\theta}} & {\sum c} & {- {\sum{s\;\theta^{2}}}} & {- {\sum{s\;\theta}}} & {- {\sum s}} \\{\sum{c\;\theta^{2}}} & {\sum{c^{2}\theta^{4}}} & {\sum{c^{2}\theta^{3}}} & {\sum{c^{2}\theta^{2}}} & {- {\sum{{cs}\;\theta^{4}}}} & {- {\sum{{cs}\;\theta^{3}}}} & {- {\sum{{cs}\;\theta^{2}}}} \\{\sum{c\;\theta}} & {\sum{c^{2}\theta^{3}}} & {\sum{c^{2}\theta^{2}}} & {\sum{c^{2}\theta}} & {- {\sum{{cs}\;\theta^{3}}}} & {- {\sum{{cs}\;\theta^{2}}}} & {- {\sum{{cs}\;\theta}}} \\{\sum c} & {\sum{c^{2}\theta^{2}}} & {\sum{c^{2}\theta}} & {\sum c^{2}} & {- {\sum{{cs}\;\theta^{2}}}} & {- {\sum{{cs}\;\theta}}} & {- {\sum{cs}}} \\{\sum{s\;\theta^{2}}} & {\sum{{cs}\;\theta^{4}}} & {\sum{{cs}\;\theta^{3}}} & {\sum{{cs}\;\theta^{2}}} & {- {\sum{s^{2}\theta^{4}}}} & {- {\sum{s^{2}\theta^{3}}}} & {- {\sum{s^{2}\theta^{2}}}} \\{\sum{s\;\theta}} & {\sum{{cs}\;\theta^{3}}} & {\sum{{cs}\;\theta^{2}}} & {\sum{{cs}\;\theta}} & {- {\sum{s^{2}\theta^{3}}}} & {- {\sum{s^{2}\theta^{2}}}} & {- {\sum{s^{2}\theta}}} \\{\sum s} & {\sum{{cs}\;\theta^{2}}} & {\sum{{cs}\;\theta}} & {\sum{cs}} & {- {\sum{s^{2}\theta^{2}}}} & {- {\sum{s^{2}\theta}}} & {- {\sum s^{2}}}\end{bmatrix}} & \lbrack 11\rbrack\end{matrix}$in which the sum indicates the sum over all the intensity values,s=sin(θ) and c=cos(θ), Y is given by:

$\begin{matrix}{Y = \begin{bmatrix}{\sum U} \\{\sum{{Uc}\;\theta^{2}}} \\{\sum{{Uc}\;\theta}} \\{\sum{Uc}} \\{\sum{{Us}\;\theta^{2}}} \\{\sum{{Us}\;\theta}} \\{\sum{Us}}\end{bmatrix}} & \lbrack 12\rbrack\end{matrix}$and the solution vector X is:

$\begin{matrix}{X = \begin{bmatrix}A \\a \\b \\c \\d \\e \\f\end{bmatrix}} & \lbrack 13\rbrack\end{matrix}$in which a=B cos(φ), b=C cos(φ), c=D cos(φ), d=B sin(φ), e=C sin(φ) andf=D sin(φ) are the least squares fitting parameters. A similar equationis used to determine φ from the intensity values for the 164^(th) scanposition. The offsets φ and φ can be found in several ways from thefitted parameters (e.g., by using the arctangent of the ratios of theappropriate fitted parameters: φ (or φ)=arc tan(f/c), arc tan(e/b), orarc tan(d/a)). The scan value increment ∂z_(165,164) between the164^(th) and 165^(th) scan positions is given by ∂z_(165,164)=φ−φ.

Without being bound by theory, it is believed that the parameters a-f ofthe solution vector X are not independent because there are only 5unknowns in the equations for the Ui and Vi. Reducing the number ofunknowns to 5 by incorporating constraints on the fitted parameters

$\left( {{e.g.},{\frac{d}{a} = {\frac{e}{b} = \frac{f}{c}}}} \right)$provides a nonlinear equation that can be solved using iterativetechniques to determine the offsets and scan increment. However, theleast squares approach discussed above is the simplest linear fit andcan be solved uniquely without iteration.

While a method for determining scan values for a scan position has beendescribed as including fitting a function to a relationship betweenintensity values of the scan position and information about the objectcorresponding to each intensity value, other methods for determining thescan increments can be used. For example, in some embodiments, therelationships illustrated by scatter plots 130,132 are transformed(e.g., by a one-dimensional transformation, which can be accomplished bya Fourier transformation) into a frequency domain with respect to theinformation about the spatial locations of the test object determinedfrom the scanning interferometry data and initial scan values. In someembodiments, the fit to the relationship is transformed. In otherembodiments, the intensity values themselves (e.g., information of thescatter plots) are transformed by, for example, a discrete Fouriertransform similar to that expressed in Eq. 5 in which the intervalsbetween successive intensity values along the x-axis are different. Thescan value increment ∂zj can be determined from transformed fits orscatter plots by:

$\begin{matrix}{{{\partial z_{j}} = {\cos^{- 1}\left( \frac{w \cdot v}{{w}{v}} \right)}},} & \left( {{Eq}.\mspace{14mu} 14} \right)\end{matrix}$where w is a two-element vector containing the real (e.g., cosine) andimaginary (e.g., sine) components of the fundamental frequency of theoscillations of the scatter plot of the jth data set and v is atwo-element vector containing the real (cosine) and imaginary (sine)components determined for the fundamental frequency of the oscillationsof the scatter plot of the jth−1 data set. The scan increment δzi can bedescribed as the angle between the fundamental frequencies of thescatter plots of the jth and jth−1 data sets. The method is repeated todetermine all the δzi.

The transform analysis of the scatter plots to determine the δzi can beaccomplished by, for example, using a phase shifting interferometry(PSI) algorithm. For example, a 7 or 13 data set PSI algorithm can beused. In some embodiments, the algorithm is selected to have reducedsensitivity to small contrast variations in the scatter plots.

While methods for determining a scan value from a relationship betweenintensity values for a scan position and information about the testobject location corresponding to each intensity value have beendescribed as using information at a single frequency (e.g., the phase ofthe fundamental frequency v₀ of each scatter plot), information fromother frequencies of the relationships expressed by scatter plots130,132 can be used. Typically, the other frequencies are higherharmonics (e.g., odd harmonics such as 3v₀, 5v₀, and 7v₀) of thefundamental frequency v₀ of each scatter plot.

As discussed above, the relationship between the intensity values for ascan position and the information about the test object spatiallocations (e.g., the scatter plots 130,132) is determined from thescanning interferometry data themselves (e.g., by using initial scanvalues for the scan positions to determine the information about thetest object spatial locations). Non-ideal scan mechanism motion canperturb these relationships so that the higher harmonics of thefundamental frequency v₀ of each scatter plot (e.g., the phases of thehigher harmonics) also carry information about the scan value for eachscan position. As discussed next, scan values can be determined from therelationships expressed by scatter plots 130,132 by correcting the phaseat the fundamental frequency based on the properly weighted phases ofhigher harmonics of the fundamental frequencies.

We first discuss a first order correction that includes information fromthe third harmonic (e.g., the frequency 3v₀). A two element vector Ŝ(v₀)containing the real (cosine) and imaginary (sine) components of thefundamental frequency v₀ of the relationship expressed by a scatter plotof intensities for a scan position corrected for vibrational distortionusing information from the frequency 3v₀ is given by:Ŝ(v ₀)=S(v ₀)+S(3v ₀)exp(−

{circumflex over (Φ)})  (Eq. 15)where S(v₀) is a two element vector containing the real and imaginarycomponents at the fundamental frequency v₀ resulting from thetransformation of the scatter plot into a frequency domain with respectto the information about the spatial locations of the test objectdetermined from the scanning interferometry data and scan values, S(3v₀)is a two element vector containing the real and imaginary components at3v₀ resulting from the transformation of the scatter plot, and is thephase of the scatter plot at the fundamental frequency given by thearctangent of the ratio of the imaginary and the real components ofuncorrected fundamental frequency the transformed scatter plot.

The scan value increment δzj between the jth scan position and the jth−1scan position is given by:

$\begin{matrix}{{\partial z_{j}} = {\cos^{- 1}\left\{ \frac{{Re}\left( {{\hat{S}\left( v_{0} \right)}_{j}{{\hat{S}}^{*}\left( v_{0} \right)}_{j - 1}} \right)}{{{\hat{S}\left( v_{0} \right)}_{j}}{{\hat{S}\left( v_{0} \right)}_{j - 1}}} \right\}}} & \left( {{Eq}.\mspace{14mu} 16} \right)\end{matrix}$where Ŝ(v₀)j is the corrected vector of real and imaginary components(Eq. 15) at the fundamental frequency for the jth scan position,Ŝ(v₀)_(j−1) is the corrected vector of real and imaginary components atthe fundamental frequency for the jth−1 scan position, * indicates thecomplex conjugate.

In general, the corrected two component vector can be determined byusing information from the N^(th)+1 higher harmonic by:

$\begin{matrix}{{\hat{S}\left( v_{0} \right)} = {\sum\limits_{k = 0}^{N}{{S\left( {\left\lbrack {{2k} + 1} \right\rbrack v_{0}} \right)}{\exp\left( {{- {\mathbb{i}2}}\; k\;\hat{\Phi}} \right)}}}} & \left( {{Eq}.\mspace{14mu} 17} \right)\end{matrix}$where Eq. 17 yields Eq. 15 when k=1. In some embodiments, the correctedvector Ŝ(v₀) is determined by using information from the N^(th)+1=5(e.g., 7, 9, 11 or higher) harmonic.

Because the scatter plots 130,132 contain only limited number ofoscillations (e.g., the scatter plots illustrate a variation in thephase of detected interference intensity of only about 2π), a window(e.g., a Hamming window) can be used to attenuate spurious oscillationsintroduced by transforming the scatter plots into the frequency domain.

The generally high point density of the scatter plots means that theNyquist criterion will not usually limit the correction order (the valueof N in Eq. (17)). Though the contributions from each spectral componentare automatically weighted by their power, it is good practice to limitthe highest order to the one whose power falls below the spectralbackground to minimize noise driven phase contributions. In certainsituations, vibrations with amplitudes of 10 nanometers or less can becorrected with a 2^(nd) order correction.

A scan value increment determined from scanning interferometry data forsuccessive scan positions may depend on the mean interference frequencyacross the modulation envelope (e.g., beneath envelope B′ of Eqs. 7 and8). Without being bound by theory it is believed that the meaninterference frequency may vary beneath the envelope resulting from reallight sources. This variation can produce small errors in the scan valueincrements (e.g., errors of no more than about 5% of the scan valueincrement) when using R greater than about 10. These errors can bemeasured and corrected. For example, scan value increments determiningpixels satisfying different values of R can be compared to determine thedependence of the scan value increment on the position of the selectedintensity values with respect to the maximum of the interference signal.This dependence can be determined using the test object itself or acalibration object.

While Eqs. 15 and 17 for determining corrected cosine and sinecomponents have been described as being applied to intensity valuesobtained from a common scan position, such corrected components can bedetermined for intensity values obtained from multiple scan positionsalong the scan dimension axis. For example, Eq. 19 can be applied tointerference signals corresponding to different test object spatiallocations, such as interference signal 75 of FIG. 2. Typically, theintensity values are obtained for scan positions separated by aboutλ₀/10 or less (e.g., by about λ₀/12 or less, by about λ₀/15 or less,e.g., by about λ₀/18 or less) along the scan dimension axis. Suchsampling densities limit (e.g., prevent) spectral overlap between thefundamental frequency of the interference signal fringes and harmonics.Applying Eqs. 15 or 17 to interference signals rather than tointensities from a common scan position advantageously determines thescan value increment for each pixel independently. Hence, the method cancorrect for scan position errors that vary for the pixels of a commonscan position. Such errors can arise, for example, from turbulence inlarge aperture interferometers.

The method of determining scan value increments from scanninginterferometry data can include correcting intensity variationsunrelated to interference intensity variations for different spatiallocations for a common scan position. For example, intensity variationscaused by test object reflectivity variations or intensity variations inthe illuminating light can be corrected by, for example, offsets and/orgain corrections of the intensity data as a function of position withinthe two-dimensional array of intensity values.

While methods for determining a scan increment between scan positionshave been described as determining a single scan value increment, morethan one (e.g., at least 3, at least 4) scan value increments can bedetermined based on intensity values for each of respective common scanpositions. For example, the two-dimensional array of intensity values(e.g., pixels) can be segmented into multiple sections and a separatescan increment determined for each section. Information about the testobject spatial locations corresponding to the intensity values withineach section is determined using the scan value increment for eachsection. This method can correct for spatial dependencies in the scanposition errors caused by, for example, wobble or turbulence.

While scanning interferometry data have been described as being obtainedby varying an OPD (e.g., by moving a test and/or reference object),other configurations are possible. For example, in some embodiments,scanning interferometry data are obtained by varying a wavelength ofthat light interferes at the detector. Each scan position typicallycorresponds to a different wavelength of detected interfering light(e.g., to a different central wavelength of the detected interferinglight). Each scan position increment typically corresponds to adifference in the wavelength between scan positions.

Any of the methods described above can be implemented, for example, incomputer hardware, software, or a combination of both. The methods canbe implemented in computer programs using standard programmingtechniques following the descriptions herein. Program code is applied toinput data to perform the functions described herein and generate outputinformation. The output information is applied to one or more outputdevices such as a display monitor. Each program may be implemented in ahigh level procedural or object oriented programming language tocommunicate with a computer system. However, the programs can beimplemented in assembly or machine language, if desired. In any case,the language can be a compiled or interpreted language. Moreover, theprogram can run on dedicated integrated circuits preprogrammed for thatpurpose.

Each such computer program is preferably stored on a storage medium ordevice (e.g., ROM or magnetic diskette) readable by a general or specialpurpose programmable computer, for configuring and operating thecomputer when the storage media or device is read by the computer toperform the procedures described herein. The computer program can alsoreside in cache or main memory during program execution. The analysismethod can also be implemented as a computer-readable storage medium,configured with a computer program, where the storage medium soconfigured causes a computer to operate in a specific and predefinedmanner to perform the functions described herein.

While methods for determining scan values from scanning interferometrydata have been described, scan values (e.g., scan value increments) canbe determined (e.g., measured) using data other than the scanninginterferometry data to be corrected. For example, in some embodiments,an inline sensor (e.g., an encoder or distance measuring interferometerthat detects light that has passed through the same optic as lightreflected from the test object) is used to measure the scan values forscan positions. In some embodiments, the inline sensor, measures motionof the scan mechanism (e.g., along the scan dimension axis). An inlinesensor can reduce (e.g., eliminate) any need to precalibrate or modifythe scan mechanism to have for purely linear motion. Hence, low-costscan mechanisms such as ball or roller bearing translation stages can beused.

In some embodiments, an inline sensor provides the information needed tocorrect for vibration induced errors. For example, an interferometerwith a polarization based sensor 200 as shown in FIG. 11 monitors thecavity OPD variation and provides compensation for both scan mechanismnon-uniformity and vibration induced motions when the measured scanvalues are used to determine (e.g., based on a generalizedtransformation) information about a test object 202. Sensor 200 includesa light source, outputs an incident light beam 204, a first beamsplitter 206, a focusing optic 208 (e.g., a lens system), a polarizationbeam splitter (PBS) 210, a reference object 212, a birefringent wedge214, and an array detector 216.

In use, incident light beam 204 passes through beam splitter 206 and isreceived by optic 208, which creates a converging beam that is receivedby PBS 210. A first polarization (e.g., a first linear polarization) ofthe converging beam is directed to test object 202. A secondpolarization (e.g., a linear polarization orthogonal to the firstpolarization) is directed to the reference object 212. PBS recombineslight reflected from the test and reference objects. The recombinedlight is received by optic 208 beamsplitter 206, which directs a firstportion 215 of the received light to wedge 214. A second portion 218 ofthe received light is directed to a second array detector (not shown)for obtaining scanning interferometry data.

Wedge 214 refracts the first and second polarizations to a differentextent. Consequently, the different polarizations are displaced withrespect to one another and form interference fringes on detector 216.The phase of the interference fringes is related to motion of the testand reference objects. During acquisition of scanning interferometrydata, the phase can be repeatedly monitored (e.g., by fast Fouriertransformation (FFT)) of the fringes detected by detector 216. The phasecan be used to determine a scan value for each scan position of thescanning interferometry data.

While inline sensors have been described, other configurations can beused. For example, in some embodiments the sensor is an external sensorthat measures the motion of the scanner carriage relative to the testobject or the motion of the optic that focuses light onto the testobject relative to the test object. FIG. 12 shows an external sensorsecured with respect to the focusing optic (e.g., the objective) thatfocuses light onto the test object. Which mode is used depends on wherethe sensor is mounted. Typically, the focusing optic is secured to thescan mechanism and moves as a unit. Hence, a sensor secured with respectto the focusing optic can measure the relative motion of the optic andtest object. In general, both vibration and non-ideal scan mechanismmotion can be corrected. If the monitored site is not identical to themeasured site then care must be taken that the motion of the testsurface at the monitored site is a bona-fide representation of themotion at the measurement site. If the external sensor is fixed to thefocusing optic body rather than to the scan mechanism carriage, scanmotion is not observable, and only vibration can be compensated for. Insome embodiments, the external sensor is a single-point sidecar sensor(SPSS) or a multipoint sidecar sensor (MPSS).

Referring to FIG. 13, a sensor includes a distance measuringinterferometer (DMI) having a heterodyne laser (e.g., a model ZMI2000heterodyne laser, Zygo Corporation—Middlefield, Conn.). The heterodynelaser outputs light at each of two closely spaced wavelengths that arecoherent with respect to one another and have opposite polarizations.The light is received by a PBS that directs a first portion (e.g., lightat one of the wavelengths) to a retro-reflector and a second portion(e.g., light at the other wavelength) to the test object (e.g., asurface of a flat panel display). Each portion of light passes twicethrough a respective ¼ waveplate 225 or 227. The light is at the twowavelengths is recombined and detected by a heterodyne detector (notshown). A fiber optic pickup (FOP) may be used to deliver the light tothe detector. The detector detects a beat frequency related to thefrequency difference between the light at the first wavelength and thelight at the second wavelength. The beat frequency is sensitive toDoppler shifts caused by relative motion of the flat panel surface.Hence, the beat frequency can be used to determine non-ideal motion ofthe scan mechanism.

The objective lens can be, for example, a lens doublet (e.g, with anumerical aperture of about 0.1) or a microscope objective (e.g., with anumerical aperture of about 0.8 and a magnification of about 50×). Inother embodiments, the wavelength of the interferometer source light isincreased (e.g., to between about 8 and 12 microns using a thermalsource) allowing numerical apertures of about 0.1 or less to be used.

Referring to FIG. 14, a SPSS having a “Koehler” illumination geometryallows reduced sensitivity to the depth of focus by illuminating a testobject (e.g., a flat panel surface) with a collimated light beam.Otherwise, the sensor of FIG. 14 is similar to that of FIG. 13.

In some embodiments, the scan value increments are determined by using amulti-point sensor (e.g., a MPSS). An image of the test object can beinterfered with an image from a relatively tilted reference surface toallow instant phase-based measurement of the scan value increments. Sucha sidecar sensor can be synchronized with the detector of theinterferometer so that the scan value increments determined by thesensor can be used, for example, for generalized transform analysis ofthe test object.

In embodiments, the multipoint sensor is a linear array orientedperpendicular to the fringes produced by the relative tilt between thetest and reference objects. The fringes detected by the sensor can beanalyzed by transformation (e.g., by Fourier transformation) todetermine the phase variation and scan value increment. FIG. 15 showssuch a sensor having a spatially extended source (e.g., anarrow-frequency source (e.g., a 670 nanometer laser diode) thatilluminates a slit or one-dimensional screen to provide fringes acrossthe field. Typically, the sensor interferometer arranged near OPD=zeroto minimize wavelength stability dependence and thereby reduce sourcecost. The objective lens could be a low cost doublet rather than anobjective since high quality imaging is not important. Furthermore, alow-NA may be used since post-processing need only extract the flatregions for analysis—thereby reducing sensitivity to focus. Only oneline image is required per main sensor image. With an 8 bit, 512 pixelsensor, the additional data volume and processing is negligible. Thesensor can be completely self-contained, with enough memory to hold theline images until requested by the main computer. Once the scan valueincrements are measured, they can be used to determine information aboutthe test object (e.g., by generalized transform methods describedherein).

Inline sensors that can be arranged to monitor motion of a scanmechanism are described by J. Schmit, A. Olszak, S. McDermed, “Whitelight interferometry with reference signal,” Proceedings of SPIE Vol.4777, 102-109, (2002), the contents of which is incorporated herein byreference.

EXAMPLES Example 1

An interferometer system was configured to obtain scanninginterferometry data in the presence of vibrations. A flat test objectwas mounted on a modulating piezoelectric transducer (PZT) to impartvibrations with controlled amplitude and frequency to the test object.The test object was tilted to introduce carrier fringes within a portionof each data set.

In a first study, a 1 Hz, 30 nanometer amplitude vibrational disturbancewas applied. FIG. 16 a shows the test object topography determined usingFDA analysis based on the scan positions and FIG. 16 b shows the testobject topography determined using the generalized transform methoddescribed herein based on scan position values determined using a leastsquares approach (e.g., Eqs. 7 and 8). FIG. 16 c shows the optical pathvariation as a function of scan position (e.g., data set number) alongthe scan dimension axis.

In a second study, a 3 Hz, 30 nanometer amplitude vibrationaldisturbance was applied. FIG. 17 a shows the test object topographydetermined using FDA analysis based on the scan positions and FIG. 17 bshows the test object topography determined using the generalizedtransform method described herein based on the scan position valuesdetermined from a least squares approach. FIG. 17 c shows the opticalpath variation as a function of scan position (e.g., data set number)along the scan dimension axis.

In both FIGS. 16 b and 17 b it is evident that the generalized approachbased on scan values provides a significantly more accuraterepresentation of the flat test object.

Example 2

Scanning interferometry data was obtained from an 800 nanometer sphereusing an interferometer with a 2.5× Nikon objective. The surfacetopography was determined out to 4 mrad local slope using both FDA basedon the scan positions and generalized transformation based on scan valueincrements determined from the scanning interferometry data.

FIG. 18 illustrates the variation in the surface shape (with sphericalcurvature removed) based on each of 4 different sets of scanninginterferometry data obtained from the sphere and using FDA analysis withinitial values (e.g., uncorrected values) for the scan positions. FIG.19 illustrates the variation in the surface shape (with sphericalcurvature removed) for 4 different determinations based on generalizedtransformation using the scan value increments determined from theinterferometry data by a least squares approach (Eqs. 7 and 8).

The variation in the topography determined by FDA is almost entirely dueto vibrational disturbances (FIG. 18). The similarity of thedeterminations using generalized transform illustrates the efficacy ofthis technique in determining information about test objects in thepresence of vibrations (FIG. 19).

Referring to FIG. 20, the non-uniformity (e.g., non-linearity) of thescan value increments determined from the scanning interferometry dataof FIGS. 18 and 19 shows both random and systematic variation along thescan dimension axis.

FIG. 21 is a flow diagram of an example process 300 for determining oneor more scan values for one or more of the multiple scan positions. Theprocess 300 includes providing scanning interferometry data for a testobject, the data including an intensity value corresponding to each ofdifferent spatial locations of the test object for each of multiple scanpositions (302). The process 300 includes determining informationrelated to the spatial locations of the test object based on thescanning interferometry data and initial scan values for each of thescan positions (304). The process 300 includes determining a scan valuefor each of the multiple scan positions based on a relationship betweenthe intensities of that scan position and the information related to thespatial locations to which the intensities of that scan positioncorrespond (306). The process 300 includes determining information aboutthe test object based on the scanning interferometry data and the scanvalues (308).

FIG. 22 is a flow diagram of an example process 320 for determining scanvalue increments between pairs of scan positions and information about atest object. The process 320 includes providing scanning interferometrydata for the test object, the data including an intensity valuecorresponding to each of different spatial locations of the test objectfor each of multiple scan positions (322). The process 320 includesdetermining information related to the spatial locations of the testobject based on the scanning interferometry data (324). The process 320includes determining a scan value increment between a pair of the scanpositions based on (a) a relationship between intensities of a firstscan position of the pair and the information related to the spatiallocations to which the intensities of the first scan position correspondand (b) a relationship between the intensities of a second scan positionof the pair and the information related to the spatial locations towhich the intensities of the second scan position correspond (326). Theprocess 320 includes determining scan value increments between pairs ofsuccessive scan positions of the scanning interferometry data (328). Theprocess 320 includes determining information about the test object basedon the scanning interferometry data and the scan value increments (330).

Other aspects, features, and advantages are within the scope of theinvention.

1. A method comprising: providing scanning interferometry data for atest object, the data comprising an intensity value corresponding toeach of different spatial locations of the test object for each ofmultiple scan positions; determining information related to the spatiallocations of the test object based on the scanning interferometry data;and determining a scan value for one of the multiple scan positionsbased on a relationship between the intensities of that scan positionand the information related to the spatial locations to which theintensities of that scan position correspond, including determining aphase of a frequency of an oscillation of the intensities of the scanposition with respect to the information related to the spatiallocations to which the intensities of the scan position correspond. 2.The method of claim 1, wherein determining information related to thespatial locations of the test object is farther based on an initial scanvalue for each of the scan positions.
 3. The method of claim 1, wherein:the scan position for which the scan value is determined is a first scanposition spaced apart from a second scan position by a scan positionincrement; and determining a scan value for the first scan positioncomprises determining a scan value increment for the scan positionincrement based on (a) the relationship between the intensities of thefirst scan position and the information related to the spatial locationsto which the intensities of the first scan position correspond and (b) arelationship between the intensities of the second scan position and theinformation related to the spatial locations to which the intensities ofthe second scan position correspond.
 4. The method of claim 1, furthercomprising determining a scan value for each of additional ones of themultiple scan positions.
 5. The method of claim 1, wherein determiningthe scan value for the scan position comprises transforming theintensities of the scan position into a frequency domain with respect tothe information related to the spatial locations to which theintensities of the scan position correspond.
 6. The method of claim 5,wherein the transformation is a one-dimensional transformation.
 7. Themethod of claim 1, wherein determining a phase of a frequency of anoscillation of the intensities of the scan position with respect to theinformation related to the spatial locations to which the intensities ofthe scan position correspond comprises determining a phase of afundamental frequency of an oscillation of the intensities of the scanposition with respect to the information related to the spatiallocations to which the intensities of the scan position correspond. 8.The method of claim 7, wherein determining the phase of the fundamentalfrequency includes using information about a phase of a frequency of atleast a second oscillation of the intensities of the scan position withrespect to the information related to the spatial locations to which theintensities of the scan position correspond, wherein the frequency ofthe second oscillation is at least 3 times the fundamental frequency. 9.A method comprising: providing scanning interferometry data for a testobject, the data comprising an intensity value corresponding to each ofdifferent spatial locations of the test object for each of multiple scanpositions; determining information related to the spatial locations ofthe test object based on the scanning interferometry data; determining ascan value increment between a first pair of the scan positions based on(a) a relationship between intensities of a first scan position of thepair and the information related to the spatial locations to which theintensities of the first scan position correspond and (b) a relationshipbetween the intensities of a second scan position of the pair and theinformation related to the spatial locations to which the intensities ofthe second scan position correspond; and repeating the determining ascan value increment to determine a scan value increment between otherpairs of the scan positions, wherein the scan value increments arenon-uniform.
 10. The method of claim 9, further comprising determiningscan value increments between all pairs of successive scan positions ofthe scanning interferometry data.
 11. The method of claim 9, comprisingdetermining information about the test object based on the scanninginterferometry data and the scan value increments.
 12. The method ofclaim 9, wherein the information is determined based on the scanninginterferometry data and initial scan values for the scan positions. 13.The method of claim 9, wherein determining a scan value incrementcomprises: fitting a first function to the at least some intensities ofthe first one of the scan positions and the information related to thespatial locations corresponding to the at least some intensities;fitting a second function to the at least some intensities of the secondone of the scan positions and the information related to the spatiallocations corresponding to the at least some intensities; anddetermining the scan increment based on fitted parameters of the firstand second functions.
 14. The method of claim 9, wherein determining ascan increment comprises: transforming at least some of the intensityvalues of the first scan position into a frequency domain with respectto the information related to the spatial locations corresponding tothose intensity values of the first scan position; and transforming atleast some of the intensity values of the second scan position into afrequency domain with respect to the information related to the spatiallocations corresponding to those intensity values of the second scanposition.
 15. The method of claim 9, wherein the determining a scanvalue increment between a pair of the scan positions comprises:determining an offset between (a) a fundamental frequency of anoscillation of the at least some intensities of the first one of thescan positions with respect to the information related to the spatiallocations corresponding to the at least some intensity values of thefirst one of the scan positions and (b) a fundamental frequency of anoscillation of the at least some intensities of the second one of thescan positions with respect to the information related to the spatiallocations corresponding to the intensity values of the second one of thescan positions.
 16. The method of claim 15, wherein the determining theoffset comprises: including information from an oscillation of at leastabout 3 times the fundamental frequency of the at least some intensitiesof the first one of the scan positions with respect to the informationrelated to the spatial locations corresponding to the at least someintensity values of the first one of the scan positions; and includinginformation from an oscillation of about 3 times the fundamentalfrequency of the at least some intensities of the first one of the scanpositions with respect to the information related to the spatiallocations corresponding to the at least some intensity values of thesecond one of the scan positions.
 17. A system comprising: aninterferometer configured to provide scanning interferometry data for atest object, the data comprising an intensity value corresponding toeach of different spatial locations of the test object for each ofmultiple scan positions; a processor configured to: determineinformation related to the spatial locations of the test object based onthe scanning interferometry data; and determine a scan value for a scanposition based on a relationship between the intensities of that scanposition and the information related to the spatial locations to whichthe intensities of that scan position correspond, including determininga phase of a frequency of an oscillation of the intensities of the scanposition with respect to the information related to the spatiallocations to which the intensities of the scan position correspond. 18.The system of claim 17, wherein the processor is configured to determineinformation about the test object based on information related to thescan values.
 19. The system of claim 17, wherein the processor isconfigured to determine the information related to the spatial locationsof the test object based on the scanning interferometry data and initialscan values for the scan positions.
 20. The system of claim 17 whereindetermining a phase of a frequency of an oscillation of the intensitiesof the scan position with respect to the information related to thespatial locations to which the intensities of the scan positioncorrespond comprises determining a phase of a fundamental frequency ofan oscillation of the intensities of the scan position with respect tothe information related to the spatial locations to which theintensities of the scan position correspond.
 21. The system of claim 20wherein the processor is configured to determine the phase of thefundamental frequency by using information about a phase of a secondfrequency of at least a second oscillation of the intensities of thescan position with respect to the information related to the spatiallocations to which the intensities of the scan position correspond,wherein the second frequency is at least 3 times the fundamentalfrequency.
 22. A method comprising: providing scanning interferometrydata for a test object, the data comprising an intensity valuecorresponding to each of different spatial locations of the test objectfor each of multiple scan positions; determining information related tothe spatial locations of the test object based on the scanninginterferometry data and initial scan values for each of the multiplescan positions; and determining refined scan values for each of themultiple scan positions based on a relationship between the intensitiesof the scan position and the information related to the spatiallocations to which the intensities of the scan position correspond; anddetermining information about the test object based on theinterferometry data and the refined scan values for each of the multiplescan positions.
 23. The method of claim 22 wherein the informationrelated to the spatial locations of the test object comprises at leastone of an optical path difference associated with one of the spatiallocations and a phase of interference associated with one of the spatiallocations.
 24. A system comprising: an interferometer configured toprovide scanning interferometry data for a test object, the datacomprising an intensity value corresponding to each of different spatiallocations of the test object for each of multiple scan positions; aprocessor configured to: provide scanning interferometry data for a testobject, the data comprising an intensity value corresponding to each ofdifferent spatial locations of the test object for each of multiple scanpositions; determine information related to the spatial locations of thetest object based on the scanning interferometry data; determine a scanvalue increment between a first pair of the scan positions based on (a)a relationship between intensities of a first scan position of the pairand the information related to the spatial locations to which theintensities of the first scan position correspond and (b) a relationshipbetween the intensities of a second scan position of the pair and theinformation related to the spatial locations to which the intensities ofthe second scan position correspond; and repeating the determining ascan value increment to determine a scan value increment between otherpairs of the scan positions, wherein the scan value increments arenon-uniform.
 25. The system of claim 24 wherein the processor isconfigured to determine a scan value increment between a pair of thescan positions by determining an offset between (a) a fundamentalfrequency of an oscillation of the at least some intensities of thefirst one of the scan positions with respect to the information relatedto the spatial locations corresponding to the at least some intensityvalues of the first one of the scan positions and (b) a fundamentalfrequency of an oscillation of the at least some intensities of thesecond one of the scan positions with respect to the information relatedto the spatial locations corresponding to the intensity values of thesecond one of the scan positions.
 26. The system of claim 25 wherein theprocessor is configured to determine the offset by including informationfrom an oscillation of at least about 3 times the fundamental frequencyof the at least some intensities of the first one of the scan positionswith respect to the information related to the spatial locationscorresponding to the at least some intensity values of the first one ofthe scan positions; and including information from an oscillation ofabout 3 times the fundamental frequency of the at least some intensitiesof the first one of the scan positions with respect to the informationrelated to the spatial locations corresponding to the at least someintensity values of the second one of the scan positions.